How do I set a texture as the world background in the game engine?
You could do this with a sky box, or a sky dome - the process is basically the same. In my opinion, sky domes look better than sky boxes, so I'll tell how to do that.

First step - add a sphere. Enter edit mode, and make it big. really big. (about as big as the sky) Select all faces, and click "flip normals". This will make it so you  see the faces from the inside of the sphere, not the outside. Then, add a material. name it something like, "sky dome," or, if your like me and usually don't bother renaming things, keep it as "Material," "Material.001," "Material.002," (Etc.)

Now we need a texture. It shouldn't be hard to find a skydome texture, the one I use often I got from textures.com (back when it was still cgtextures). You should be able to select the sphere, and UV unwrap it with the default sphere projection. Depending on what texture you find, you may want to delete the bottom half of the sphere before  trying to unwrap it)

Most people will tell you to set the material to "Shadeless." However, you get more control over the brightness by moving the "emission" value up and down. 0 is normal material. 1 is pretty close to what shadeless would look like, and 2 is much brighter. Plus, when using the Emission setting, lamps will effect the sky, (which you may, or may not want, actually, depending on your specific case)

The same method would apply to a cube, only you'd need a different texture, and you'd be using a cube instead of a sphere.